On special regularity properties of solutions of the Zakharov-Kuznetsov equation

We study special regularity properties of solutions to the initial value problem associated to the Zakharov-Kuznetsov equation in three dimensions. We show that the initial regularity of the data in a family of half-spaces propagates with infinite speed. By dealing with the finite envelope of a class of these half-spaces we extend the result to the complement of a family of cones in $\mathbb{R}^3$.


FELIPE LINARES AND GUSTAVO PONCE
and Regarding well-posedness for the IVP (1.1) the best local result available was obtained by Ribaud and Vento [28] for initial data in H s (R 3 ), s > 1. In [26] Molinet and Pilod proved that this local result can be extended globally in time. Previous local well-posedness results for (1.1) were established by Linares and Saut in [25].
The equation in (1.1) has a two dimensional (2D) analogous which was also derived in [30]. As in the 3D case there have been a lot of interest in the study of well-posedness in low regularity spaces for the associated IVP. The best local result available in the literature was obtained for initial data in H s (R 2 ), for s > 1/2, independently by Molinet and Pilod [26] and Grünrock and Herr [11]. Global well-posedness was proved for data in H j (R 2 ), j ∈ Z + , j ≥ 1 by Faminskii [7] and for initial data in H s (R 2 ), s ≥ 1, by Linares and Pastor [22]. The scaling argument suggests local well-posedness in the 3D case for data in H s (R 3 ) with s ≥ −1/2 and in 2D for data in H s (R 2 ) with s ≥ −1, therefore the local well-posedness results commented above are far from these values. It is an interesting open problem to establish the optimal Sobolev space where local wellposedness is attained in either case.
In the case of the so called generalized ZK equation, i.e.
∂ t u + ∂ x ∆u + u k ∂ x u = 0, k ∈ Z + , (1.3) we shall briefly described what is known for the well-posedness of the associated IVP. In the 2D case, the scale argument suggests local well-posedness results for data in H s (R 2 ), for s > s k = 1 − 2/k. Sharp local results were obtained by Ribaud and Vento [29], for k ≥ 4. In [10], Grünrock proved the local well-posedness iṅ H s k (R n ) for n = 2, 3 and s k = n/2 − 2/k, k ≥ 3. For the nonlinearity k = 2, for which L 2 (R 2 ) is the space suggested by the scaling argument, local result was shown for data in H s (R 2 ), for s ≥ 1/4 in [29]. Also for the nonlinearity k = 2 global well-posedness in H s (R 2 ), s > 53/63, for initial data with suitable L 2 norm, was established by Linares and Pastor in [23]. In the 3D case we refer to Grünrock [9] for low regularity well-posedness results regarding the equation with nonlinearity k = 2. We observe that in 2D case there exists a nonsingular linear transformation that symmetrizes the equation (1.2) (see [7,11] and references therein). More precisely, the change of variables with αµ = 0, suitable chosen takes the equation in (1.2) into the equation This in particular will allow to consider the IVP associated to (1.4) instead of the IVP (1.2) without changing the well-posedness theory.
As is well-known, energy estimates obtained by using the commutator estimates in [17] permit to establish local well-posedness results for the IVP (1.2) in Sobolev spaces H s (R 2 ), for s > 2 (see [5] for instance). One can loose (or strength) a little bit the previous local result combining the method introduced by Kenig [18] to study the IVP associated to the KP-I equation and the symmetric equation (1.4). We point out that this change of variables was used in [11] to obtain the result commented above via the Fourier restriction method.
Another interesting issue is that concerning the traveling wave solutions or solitary waves for the ZK equation and for the generalized ZK equation. In [5] de Bouard studied the existence and the orbital stability of such solutions. Recently, Côte, Muñoz, Pilod, and Simpson [6] among other results, showed the asymptotic stability of the solitons in the 2D case.
Our purpose here is the study of the propagation of regularity for solutions of the IVP (1.1). In [15] Isaza, Linares and Ponce, considering suitable solutions of the IVP associated to the generalized KdV equation established the propagation of regularity in the right hand side of the data for positive times. More precisely, 15]). If u 0 ∈ H 3/4 + (R) and for some l ∈ Z + , l ≥ 1 and then the solution of the IVP (1.5) provided by the local theory in [19] satisfies that for any v > 0 and > 0 for j = 0, 1, . . . , l with c = c(l; u 0 3/4 + ,2 ; ∂ l x u 0 L 2 ((x0,∞)) ; v; ; T ). In particular, for any t ∈ (0, T ], the restriction of u(·, t) to any interval (x 0 , ∞) belongs to H l ((x 0 , ∞)).
Moreover, for any v ≥ 0, > 0 and R > 0 ; v; ; R; T ). The property described in the Theorem 1.1 is intrinsic to suitable solutions of some nonlinear dispersive models, see for instance [13,14], where analogous results for the Kadomtsev-Petviashvili II and Benjamin-Ono equations were proved.
To state our results we need to describe the class of solutions in which it applies. Thus, we first recall a result which is direct consequence of the energy estimates obtained by combining the commutator estimates in [17], the Sobolev embedding theorem and the argument in [2]. (1.6) We observe that Also, we shall use the following result which is a consequence of the arguments given in [25].
To delineate our results we introduce some notations: for a, b, c, d, f ∈ R we define the half-space Our main result in this paper is the following :
(1.14) One notices that the fact > 0 allows us to write (C x0−vtv+ ŵ,ŵ,θ ) c as the union of finitely many half-spaces P ar,br,cr,dr−vt− /2 's, and that the condition (1.14) guarantees that the a r , b r , c r 's satisfy (1.8).
Thus, for ∼ 1 − one gets a gain of 3/8 − derivatives (in the x-variable). Notice that (roughly speaking) this gain of derivative in the x-variable extends to all variables (x, y, z) if v 0 is supported inside the cone √ 3ξ 1 > ξ 2 2 + ξ 2 3 which is similar to that described in (1.8) for the physical space.
The previous observation can also be applied to solutions of the linear problem associated to the ZK equation in 2D. More precisely, it was proved in [22] that solutions of the linear problem, U (t)f = e it(ξ(ξ 2 +η 2 )) f satisfy the Strichartz estimates. Lemma 1.9. Let 0 ≤ ε < 1/2 and 0 ≤ θ ≤ 1. Then, Remark 1.10. Our second result describes the persistence properties and regularity effects, for positive times, in solutions associated with data having polynomial decay in an appropriate half-space.
The proof of Theorem 1.11 follows an argument similar to that given below in the proof of Theorem 1.4 so it will be omitted (for details see [15]). 2. Proof of Theorem 1.4. Our starting point will be a weighted energy identity. This was motivated by the original proof of the so called Kato smoothing effect deduced in [16]. To obtain it we apply the operator ∂ α = ∂ α1 x ∂ α2 y ∂ α3 z to the equation in (1.1), multiply the result by 2∂ α uχ with χ = χ(ax+by +cz −d+vt), a, b, c, d, v ∈ R and χ, χ ≥ 0 to be chosen, and integrate the result in R 3 to get after some integrations by parts that We claim : if (1.8) holds, i.e.
then there exists > 0 such that To prove the claim we shall use the notations Hence, since Without loss of generality we shall assume from now on that d = 0. First, we shall consider the case: j = |α| = 2.
Case: j = |α| = 3. By hypothesis on the data u 0 for some a, b, c for which (2.2) holds one has For any > 0 and R > 4 let Let v > 0. Consider the identity (2.1) for all α with |α| = 3. Adding in α the previous argument combined with (2.6) provides, after integration in time, the bounds for the terms E 1 , E 3 , E 6 , E 9 . Also we have (omitting the summation in |α| = 3) by integration by parts that Thus, the first term in the r.h.s of (2.8) will be estimated by Gronwall's inequality and (1.7) and the second one, after integration in time, by (1.6) and (2.6). To bound E 10,3 we see that where we have omitted the summation sign in β, β , α with |β| = |β | = 2, |α| = 3. By integration by parts one has that For the last term in (2.10) it follows from Gagliardo-Nirenberg and Young inequalities that    | ∂ β u ∂ β u ∂ α u χd x|.